Gábor Korchmáros

Curves of large genus covered by the hermitian curve

For the Hermitian curve H defined over the finite field , we give a complete classification of Galois coverings of H of prime degree. The corresponding quotient curves turn out to be special cases of wider families of curves -covered by H arising from subgroups of the special linear group SL(2,F q ) or from subgroups in the normaliser of the Singer group of the projective unitary group . Since curves -covered by H are maximal over , our results provide some classification and existence theorems for maximal curves having large genus, as well as several values for the spectrum of the genera of maximal curves. For every q 2, both the upper limit and the second largest genus in the spectrum are known, but the determination of the third largest value is still in progress. A discussion on the “third largest genus problem“ including some new results and a detailed account of current work is given.

Embedding of a Maximal Curve in a Hermitian Variety

Let X be a projective, geometrically irreducible, non-singular, algebraic curve defined over a finite field Fq2 of order q2. If the number of Fq2-rational points of X satisfies the Hasse–Weil upper bound, then X is said to be Fq2-maximal. For a point P0 ∈ X(Fq2), let π be the morphism arising from the linear series D: = |(q + 1)P0|, and let N: = dim(D). It is known that N ≥ 2 and that π is independent of P0 whenever X is Fq2-maximal.

Note on (q+2)-Sets in A Galois Plane of Order q

Let Ω be a (q+2)-set in a projective plane PG(2, q) and let ϕ ⊆ Ω be the set of the points of Ω with the property that every line containing a point of ϕ intersects Ω in two points. In this paper we prove that: If |ϕ gt; 2, then q is even; if |ϕ > q÷2, then ϕ = Ω for each q even, there exists an Ω such that |ϕ = q÷2.

On Plane Maximal Curves

AbstractThe number N of rational points on an algebraic curve of genus g over a finite field

Algebraic curves with a large non-tame automorphism group fixing no point

{\mathbb{F}}_q

Translation Planes of Order 112

satisfies the Hasse–Weil bound

Complete arcs arising from conics

N \leqslant q + 1 + 1g\sqrt q